1. © 2002 Prentice-Hall, Inc.Chap 7-1 Statistics for Managers using Excel 3 rd Edition Chapter 7 Fundamentals of Hypothesis Testing: One-Sample Tests

2. © 2002 Prentice-Hall, Inc. Chap 7-2 Chapter Topics Hypothesis testing methodology Z test for the mean ( known) P-value approach to hypothesis testing Connection to confidence interval estimation One-tail tests T test for the mean ( unknown) Z test for the proportion Potential hypothesis-testing pitfalls and ethical considerations

3. © 2002 Prentice-Hall, Inc. Chap 7-3 What is a Hypothesis? A hypothesis is a claim (assumption) about the population parameter Examples of parameters are population mean or proportion The parameter must be identified before analysis I claim the mean GPA of this class is 3.5! © 1984-1994 T/Maker Co.

4. © 2002 Prentice-Hall, Inc. Chap 7-4 The Null Hypothesis, H 0 States the assumption (numerical) to be tested e.g.: The average number of TV sets in U.S. Homes is at least three ( ) Is always about a population parameter ( ), not about a sample statistic ( )

5. © 2002 Prentice-Hall, Inc. Chap 7-5 The Null Hypothesis, H 0 Begins with the assumption that the null hypothesis is true Similar to the notion of innocent until proven guilty Refers to the status quo Always contains the “=” sign May or may not be rejected (continued)

6. © 2002 Prentice-Hall, Inc. Chap 7-6 The Alternative Hypothesis, H 1 Is the opposite of the null hypothesis e.g.: The average number of TV sets in U.S. homes is less than 3 ( ) Challenges the status quo Never contains the “=” sign May or may not be accepted Is generally the hypothesis that is believed (or needed to be proven) to be true by the researcher

7. © 2002 Prentice-Hall, Inc. Chap 7-7 Hypothesis Testing Process Identify the Population Assume the population mean age is 50. ( ) REJECT Take a Sample Null Hypothesis No, not likely!

8. © 2002 Prentice-Hall, Inc. Chap 7-8 Sampling Distribution of = 50 It is unlikely that we would get a sample mean of this value...... Therefore, we reject the null hypothesis that m = 50. Reason for Rejecting H 0  20 If H 0 is true... if in fact this were the population mean.

9. © 2002 Prentice-Hall, Inc. Chap 7-9 Level of Significance, Defines unlikely values of sample statistic if null hypothesis is true Called rejection region of the sampling distribution Is designated by, (level of significance) Typical values are.01,.05,.10 Is selected by the researcher at the beginning Provides the critical value(s) of the test

10. © 2002 Prentice-Hall, Inc. Chap 7-10 Level of Significance and the Rejection Region H 0 :   3 H 1 :  < 3 0 0 0 H 0 :   3 H 1 :  > 3 H 0 :   3 H 1 :   3    /2 Critical Value(s) Rejection Regions

11. © 2002 Prentice-Hall, Inc. Chap 7-11 Errors in Making Decisions Type I Error Rejects a true null hypothesis Has serious consequences The probability of Type I Error is Called level of significance Set by researcher Type II Error Fails to reject a false null hypothesis The probability of Type II Error is The power of the test is

12. © 2002 Prentice-Hall, Inc. Chap 7-12 Errors in Making Decisions Probability of not making Type I Error Called the confidence coefficient (continued)

13. © 2002 Prentice-Hall, Inc. Chap 7-13 Result Probabilities H 0 : Innocent The Truth Verdict InnocentGuilty Decision H 0 TrueH 0 False Innocent CorrectError Do Not Reject H 0 1 -  Type II Error (  ) Guilty Error Correct Reject H 0 Type I Error (  ) Power (1 -  ) Jury Trial Hypothesis Test

14. © 2002 Prentice-Hall, Inc. Chap 7-14 Type I & II Errors Have an Inverse Relationship   If you reduce the probability of one error, the other one increases so that everything else is unchanged.

15. © 2002 Prentice-Hall, Inc. Chap 7-15 Factors Affecting Type II Error True value of population parameter Increases when the difference between hypothesized parameter and its true value decrease Significance level Increases when decreases Population standard deviation Increases when increases Sample size Increases when n decreases n

16. © 2002 Prentice-Hall, Inc. Chap 7-16 How to Choose between Type I and Type II Errors Choice depends on the cost of the errors Choose smaller Type I Error when the cost of rejecting the maintained hypothesis is high A criminal trial: convicting an innocent person The Exxon Valdez: causing an oil tanker to sink Choose larger Type I Error when you have an interest in changing the status quo A decision in a startup company about a new piece of software A decision about unequal pay for a covered group

17. © 2002 Prentice-Hall, Inc. Chap 7-17 Critical Values Approach to Testing Convert sample statistic (e.g.: ) to test statistic (e.g.: Z, t or F –statistic) Obtain critical value(s) for a specified from a table or computer If the test statistic falls in the critical region, reject H 0 Otherwise do not reject H 0

18. © 2002 Prentice-Hall, Inc. Chap 7-18 p-Value Approach to Testing Convert Sample Statistic (e.g. ) to Test Statistic (e.g. Z, t or F –statistic) Obtain the p-value from a table or computer p-value: Probability of obtaining a test statistic more extreme ( or ) than the observed sample value given H 0 is true Called observed level of significance Smallest value of that an H 0 can be rejected Compare the p-value with If p-value, do not reject H 0 If p-value, reject H 0

19. © 2002 Prentice-Hall, Inc. Chap 7-19 General Steps in Hypothesis Testing e.g.: Test the assumption that the true mean number of of TV sets in U.S. homes is at least three ( Known) 1.State the H 0 2.State the H 1 3.Choose 4.Choose n 5.Choose Test

20. © 2002 Prentice-Hall, Inc. Chap 7-20 100 households surveyed Computed test stat =-2, p-value =.0228 Reject null hypothesis The true mean number of TV sets is less than 3 (continued) Reject H 0  -1.645 Z 6.Set up critical value(s) 7. Collect data 8. Compute test statistic and p-value 9. Make statistical decision 10. Express conclusion General Steps in Hypothesis Testing

21. © 2002 Prentice-Hall, Inc. Chap 7-21 One-tail Z Test for Mean ( Known) Assumptions Population is normally distributed If not normal, requires large samples Null hypothesis has or sign only Z test statistic

22. © 2002 Prentice-Hall, Inc. Chap 7-22 Rejection Region Z 0 Reject H 0 Z 0 H 0 :  0 H 1 :  <  0 H 0 :  0 H 1 :  >  0 Z Must Be Significantly Below 0 to reject H 0 Small values of Z don’t contradict H 0 Don’t Reject H 0 !

23. © 2002 Prentice-Hall, Inc. Chap 7-23 Example: One Tail Test Q. Does an average box of cereal contain more than 368 grams of cereal? A random sample of 25 boxes showed = 372.5. The company has specified  to be 15 grams. Test at the  0.05 level. 368 gm. H 0 :  368 H 1 :  > 368

24. © 2002 Prentice-Hall, Inc. Chap 7-24 Finding Critical Value: One Tail Z.04.06 1.6.9495.9505.9515 1.7.9591.9599.9608 1.8.9671.9678.9686.9738.9750 Z 0 1.645. 05 1.9.9744 Standardized Cumulative Normal Distribution Table (Portion) What is Z given  = 0.05?  =.05 Critical Value = 1.645.95

25. © 2002 Prentice-Hall, Inc. Chap 7-25 Example Solution: One Tail Test  = 0.5 n = 25 Critical Value: 1.645 Test Statistic: Decision: Conclusion: Do Not Reject at  =.05 No evidence that true mean is more than 368 Z 0 1.645.05 Reject H 0 :  368 H 1 :  > 368 1.50

26. © 2002 Prentice-Hall, Inc. Chap 7-26 p -Value Solution Z 0 1.50 P-Value =.0668 Z Value of Sample Statistic From Z Table: Lookup 1.50 to Obtain.9332 Use the alternative hypothesis to find the direction of the rejection region. 1.0000 -.9332.0668 p-Value is P(Z  1.50) = 0.0668

27. © 2002 Prentice-Hall, Inc. Chap 7-27 p -Value Solution (continued) 0 1.50 Z Reject (p-Value = 0.0668)  (  = 0.05) Do Not Reject. p Value = 0.0668  = 0.05 Test Statistic 1.50 is in the Do Not Reject Region 1.645

28. © 2002 Prentice-Hall, Inc. Chap 7-28 One-tail Z Test for Mean ( Known) in PHStat PHStat | one-sample tests | Z test for the mean, sigma known … Example in excel spreadsheet

29. © 2002 Prentice-Hall, Inc. Chap 7-29 Example: Two-Tail Test Q. Does an average box of cereal contain 368 grams of cereal? A random sample of 25 boxes showed = 372.5. The company has specified  to be 15 grams. Test at the  0.05 level. 368 gm. H 0 :  368 H 1 :  368

30. © 2002 Prentice-Hall, Inc. Chap 7-30  = 0.05 n = 25 Critical Value: ±1.96 Example Solution: Two-Tail Test Test Statistic: Decision: Conclusion: Do Not Reject at  =.05 No Evidence that True Mean is Not 368 Z 0 1.96.025 Reject -1.96.025 H 0 :  368 H 1 :  368 1.50

31. © 2002 Prentice-Hall, Inc. Chap 7-31 p-Value Solution (p Value = 0.1336)  (  = 0.05) Do Not Reject. 0 1.50 Z Reject  = 0.05 1.96 p Value = 2 x 0.0668 Test Statistic 1.50 is in the Do Not Reject Region Reject

32. © 2002 Prentice-Hall, Inc. Chap 7-32 PHStat | one-sample tests | Z test for the mean, sigma known … Example in excel spreadsheet Two-tail Z Test for Mean ( Known) in PHStat

33. © 2002 Prentice-Hall, Inc. Chap 7-33 Connection to Confidence Intervals

34. © 2002 Prentice-Hall, Inc. Chap 7-34 t Test: Unknown Assumption Population is normally distributed If not normal, requires a large sample T test statistic with n-1 degrees of freedom

35. © 2002 Prentice-Hall, Inc. Chap 7-35 Example: One-Tail t Test Does an average box of cereal contain more than 368 grams of cereal? A random sample of 36 boxes showed X = 372.5, and  s  15. Test at the  0.01 level. 368 gm. H 0 :  368 H 1 :  368  is not given

36. © 2002 Prentice-Hall, Inc. Chap 7-36 Example Solution: One-Tail  = 0.01 n = 36, df = 35 Critical Value: 2.4377 Test Statistic: Decision: Conclusion: Do Not Reject at  =.01 No evidence that true mean is more than 368 t 35 0 2.437 7.01 Reject H 0 :  368 H 1 :  368 1.80

37. © 2002 Prentice-Hall, Inc. Chap 7-37 p -Value Solution 0 1.80 t 35 Reject (p Value is between.025 and.05)  (  = 0.01). Do Not Reject. p Value = [.025,.05]  = 0.01 Test Statistic 1.80 is in the Do Not Reject Region 2.4377

38. © 2002 Prentice-Hall, Inc. Chap 7-38 PHStat | one-sample tests | t test for the mean, sigma known … Example in excel spreadsheet t Test: Unknown in PHStat

39. © 2002 Prentice-Hall, Inc. Chap 7-39 Proportion Involves categorical values Two possible outcomes “Success” (possesses a certain characteristic) and “Failure” (does not possesses a certain characteristic) Fraction or proportion of population in the “success” category is denoted by p

40. © 2002 Prentice-Hall, Inc. Chap 7-40 Proportion Sample proportion in the success category is denoted by p S When both np and n(1-p) are at least 5, p S can be approximated by a normal distribution with mean and standard deviation (continued)

41. © 2002 Prentice-Hall, Inc. Chap 7-41 Example: Z Test for Proportion Q. A marketing company claims that it receives 4% responses from its mailing. To test this claim, a random sample of 500 were surveyed with 25 responses. Test at the  =.05 significance level.

42. © 2002 Prentice-Hall, Inc. Chap 7-42 Z Test for Proportion: Solution  =.05 n = 500 Do not reject at  =.05 H 0 : p .04 H 1 : p .04 Critical Values:  1.96 Test Statistic: Decision: Conclusion: Z 0 Reject.025 1.96-1.96 1.14 We do not have sufficient evidence to reject the company’s claim of 4% response rate.

43. © 2002 Prentice-Hall, Inc. Chap 7-43 p -Value Solution (p Value = 0.2542)  (  = 0.05). Do Not Reject. 0 1.14 Z Reject  = 0.05 1.96 p Value = 2 x.1271 Test Statistic 1.14 is in the Do Not Reject Region Reject

44. © 2002 Prentice-Hall, Inc. Chap 7-44 Z Test for Proportion in PHStat PHStat | one-sample tests | z test for the proportion … Example in excel spreadsheet

45. © 2002 Prentice-Hall, Inc. Chap 7-45 Potential Pitfalls and Ethical Considerations Randomize data collection method to reduce selection biases Do not manipulate the treatment of human subjects without informed consent Do not employ “data snooping” to choose between one-tail and two-tail test, or to determine the level of significance

46. © 2002 Prentice-Hall, Inc. Chap 7-46 Potential Pitfalls and Ethical Considerations Do not practice “data cleansing” to hide observations that do not support a stated hypothesis Report all pertinent findings (continued)

47. © 2002 Prentice-Hall, Inc. Chap 7-47 Chapter Summary Addressed hypothesis testing methodology Performed Z Test for the mean ( Known) Discussed p –Value approach to hypothesis testing Made connection to confidence interval estimation

48. © 2002 Prentice-Hall, Inc. Chap 7-48 Chapter Summary Performed one-tail and two-tail tests Performed t test for the mean ( unknown) Performed Z test for the proportion Discussed potential pitfalls and ethical considerations (continued)